Effective module lattices and their shortest vectors
Abstract
We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.
Keywords
Cite
@article{arxiv.2402.10305,
title = {Effective module lattices and their shortest vectors},
author = {Nihar Gargava and Vlad Serban and Maryna Viazovska and Ilaria Viglino},
journal= {arXiv preprint arXiv:2402.10305},
year = {2025}
}
Comments
21 pages. This paper has now been broken into two papers; namely 2510.12893 and 2510.11673. The former improves Theorem 3 from this preprint with the required rank condition t>26 improved to t>10. The latter is a rewrite of the combinatorics part of the paper about lifts of codes and includes a proof of Theorem 4 from this preprint. No further changes will be made to this preprint